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Double temperaments?

🔗Daniel A. Wier <dawiertx@sbcglobal.net>

6/10/2006 3:12:46 AM

(I could've sworn I asked this question here years ago... maybe not.)

The current discussion on slendro and pelog got me thinking somehow: is there a word for a linear temperament involving a fifth/generator lowered by a certain amount and an octave/period raised by the same interval? Or the fifth raised and the octave lowered?

I want to say "double temperament", which I Googled for, but found no pertinent results.

Bohlen-Pierce is an example, if understood as having a fifth of 32/21 (raised by a septimal comma) and an octave of 64/33 (lowered by the same). Another is 49 equal divisions of a twelfth which I've proposed as an alternative to 31-tET; it has a slightly better fifth but a slightly worse fourth.

I also came up with a 9-tone scale with a fifth of 40/27 (lowered by a syntonic comma) and an octave of 81/40 (raised by the same):

0: 0.000 (1/1)
1: 139.391 (64000/59049)
2: 278.782
3: 401.666 (129140163/102400000)
4: 541.058 (2187/1600)
5: 680.449 (40/27)
6: 819.840 (2560000/1594323)
7: 942.724
8: 1082.115 (4782969/2560000)
9: 1221.506 (81/40)

... a bad "pelog" tuning. This is also close to 14 equal divisions of a twelfth, or 53-tET using every sixth note. A 13-limit JI version is 1/1 13/12 7/6 33/26 11/8 40/27 8/5 26/15 15/8 81/40. (14/11 could be used in place of 33/26, but it deviates from ET a bit much.)

~Danny~